On the canonical connection for smooth envelopes

TL;DR

This paper explores the concept of a canonical connection in the context of smooth envelopes, demonstrating the existence of such a connection in a specific algebra, which is crucial for differential calculus on generalized smooth manifolds.

Contribution

It provides an example of an algebra that always admits a canonical connection, addressing a key question in the theory of smooth envelopes.

Findings

Existence of a canonical connection in a specific algebra.

Clarification of the relationship between derivations and smooth envelopes.

Implications for differential calculus on generalized manifolds.

Abstract

A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the vice-versa can be done as well. In a physical language, this would corresponds to the existence of a canonical connection. In this paper we show an example of an algebra which always possesses such a connection.